Questions: Tarski's Undefinability Theorem and Truth

Question 1

Gödel numbering allows arithmetic to encode statements about its own syntax. Tarski's theorem says arithmetic cannot do something analogous for semantics. What exactly is ruled out?

• A. Arithmetic cannot prove its own consistency
• B. Arithmetic cannot express any self-referential sentence
• C. Arithmetic cannot define a formula Truth(x) that correctly identifies which sentences are true in the standard model
• D. Arithmetic cannot quantify over all natural numbers simultaneously

Answer: (C)

Tarski's undefinability theorem rules out the existence of a formula Truth(x) in the language of arithmetic such that Truth(⌜φ⌝) ↔ φ holds for all sentences φ. Arithmetic can perfectly well encode syntactic facts (which strings are valid formulas, what the Gödel code of a formula is, whether one formula is a substitution instance of another) — these are recursive relations on natural numbers. What it cannot do is define what those formulas mean: it cannot express, from the inside, which of its sentences are actually true in the standard model. Option A describes Gödel's second incompleteness theorem; option B is false (the diagonal lemma constructs self-referential sentences); option D is wrong (bounded and unbounded quantification is standard in arithmetic).

Question 2

Tarski's theorem shows that truth in a model can seldom be rigorously defined — not even in a metalanguage.

• T. True
• F. False

Answer: False

This is a critical misreading of the theorem. Tarski's theorem shows that truth in a model cannot be defined *within the object language itself*. It says nothing against defining truth from outside. Indeed, the standard Tarski semantics (M ⊨ φ defined by induction on formula structure) is a perfectly rigorous definition of truth — it just lives in the metalanguage. The theorem's lesson is not that truth is indefinable, but that it requires a richer language to define it than the language whose truth is being defined. A metalanguage can define object-language truth; the object language cannot define its own truth.

Question 3

The Tarski hierarchy of metalanguages provides a single unified truth predicate that covers sentences at most levels.

• T. True
• F. False

Answer: False

The Tarski hierarchy does the opposite: it generates an infinite regress rather than a unified predicate. Truth for level-n sentences is defined in a level-(n+1) metalanguage. But then that metalanguage's own truth requires a level-(n+2) language, and so on indefinitely. There is no level at which you can capture truth for all levels simultaneously — no 'super-language' that collapses the hierarchy. This infinite regress is a fundamental consequence of the undefinability theorem, not a defect to be engineered away. Tarski himself was clear that this prevents any language from having a single, self-contained truth predicate.

Question 4

The Tarski hierarchy implies that there is no single unified truth predicate that covers sentences at all levels of the hierarchy.

• T. True
• F. False

Answer: True

Tarski's theorem forces truth to be language-relative: truth for level-n sentences lives in a level-(n+1) metalanguage. Attempting to define a universal truth predicate collapses into the Liar paradox (via the diagonal lemma), as the proof shows. Any candidate for a unified truth predicate would allow construction of a self-referential Liar sentence, generating contradiction. The hierarchy is not optional scaffolding but a structural necessity imposed by the theorem.

Question 5

What is the crucial distinction between arithmetic's ability to represent its own syntax and its inability to represent its own semantics, and why does this distinction matter?

Model answer: Syntax concerns the formal structure of expressions — which strings are formulas, what their Gödel codes are, whether one formula is a substitution instance of another. These are all recursive (computable) functions on natural numbers, and arithmetic can define them. Semantics concerns what formulas mean — whether they are true in the standard model. Truth is not a recursive predicate; it cannot be captured by any arithmetical formula. The distinction matters because self-reference through syntax (via Gödel coding) is available in arithmetic and enables Gödel's incompleteness theorems, while self-reference through semantics is blocked by Tarski's theorem. Arithmetic can 'talk about' its own proofs (a syntactic notion) but cannot 'talk about' its own truth (a semantic notion). This explains why Gödel produced incompleteness results (using provability, not truth) while Tarski showed truth is irreducibly metalinguistic.

The syntax/semantics divide is one of the deepest distinctions in logic. Syntax is algorithmic — it can be mechanically checked and encoded. Semantics is interpretive — it requires a model, a structure, an interpretation that stands outside the formal system. Tarski's theorem makes this informal distinction into a mathematical theorem: the gap between syntax and semantics is not closable from within a sufficiently rich language.